How should Buddhists deal with Loneliness

This seems like a two-folded question to me.

People who experience difficulties in relationship to other people can learn to break out of it, how difficult it may seem. It is a type of training where buddhism can help to a certain degree. However, there are a lot of situations in our modern lives, that calls for - shall we say - modern skill building, in addition to buddhist practice.

Dealing with the pain associated with the above is - on the other hand - very much a topic to be dealt with the scope of buddhism. It is likely open to interpretation, but loneliness could arguably be described in different ways, dependent on the situation:

Loneliness could be seen as painful in and of itself (dukkha dukkha). It is not a misconception that loneliness hurts, just as there is no misconception that a knife wound hurts, and therefore it is not necessarily caused by avijja. In this case, we may only be able to deal with the pain by stilling it temporarily through samatha in our worst moments.

It is also a situation that calls for other interventions to break out of loneliness, just like a knife wound requires going to an ER. Generally speaking, it could mean the developing of new habits for activation, establishing contact with others, or developing social skills for those who need it. These are examples of skillbuilding outside the scope of buddhism, and i don't believe it would be helpful to simply dismiss the above as tanha/upadana. Buddhism has never been about subjecting ourselves to unnecessary suffering, provided that we can prevent it.

However, loneliness could also make us feel unsatisfied because we cling to involuntary false assumptions regarding ourselves and relationships (attavadupadana). For instance, a person who lacks the relationship he/she believes he/she deserves or needs. This is a trickier situation, since it is conditioned by avijja, tanha or upadana to a greater degree than the previous example.

Using another strictly hypothetical example, if we feel unsatisfied by our loneliness because we have a strong belief that every person on the planet must like us, or that we are fundamentally impossible to be appreciated by others, it becomes another examples created by attavadupadana.

Another hypothetical example: if we believe that a happy life can only be lived with no loneliness at all, we are bound to be unhappy as soon as we aren't surrounding ourselves with people at every price (vibhavatanha).

These are some examples where one could benefit from buddhist insights regarding unhealthy expectations, desire or clinging, and consequently practicing nekkhamma for renunciating unbeneficial assumptions.

These are complicated issues, and dealing with them requires more than the Q&A format here. Also, these things doesn't rule each other out, and a way to less suffering could possibly consist of a combination of several things.

Please bear in mind that there are a lot of people struggling with this, and not everyone is willing to openly admit it. To me, the latter is a rather big step in a new direction, hopefully to a less painful life.

Is companionship valued in Buddhism?

Yes, but only noble companionship. It's not just half of the holy life, but it is the entire holy life i.e. the journey to enlightenment.

From SN 45.2:

Then the Venerable Ānanda approached the Blessed One. Having approached, he paid homage to the Blessed One, sat down to one side, and said to him:

“Venerable sir, this is half of the holy life, that is, good friendship, good companionship, good comradeship.”

“Not so, Ānanda! Not so, Ānanda! This is the entire holy life, Ānanda, that is, good friendship, good companionship, good comradeship. When a bhikkhu has a good friend, a good companion, a good comrade, it is to be expected that he will develop and cultivate the Noble Eightfold Path.

“And how, Ānanda, does a bhikkhu who has a good friend, a good companion, a good comrade, develop and cultivate the Noble Eightfold Path? Here, Ānanda, a bhikkhu develops right view, which is based upon seclusion, dispassion, and cessation, maturing in release. He develops right intention … right speech … right action … right livelihood … right effort … right mindfulness … right concentration, which is based upon seclusion, dispassion, and cessation, maturing in release. It is in this way, Ānanda, that a bhikkhu who has a good friend, a good companion, a good comrade, develops and cultivates the Noble Eightfold Path.

Furthermore, you should choose your friends carefully according to AN 7.36:

A friend gives what is hard to give,
and does what’s hard to do.
They put up with your harsh words,
and with things hard to endure.

They tell you their secrets,
and keep your secrets for you.
They don’t abandon you in times of trouble,
or look down on you in times of loss.

The person in whom
these things are found is your friend.
If you want to have a friend,
you should keep company with such a person.”

And AN 7.37:

They’re lovable, respected, and admired,
an admonisher who accepts admonishment,
speaks on deep matters,
and doesn’t urge you to do bad.

The person in whom
these things are found is your friend.
If you want to have a friend,
benevolent and compassionate,
you should keep company with such a person,
even if they send you away.”

On seclusion and pursuing the holy life alone, the famous sutta is the Rhinoceros Sutta (Sutta Nipata 1.3) which I quote below.

All of the following apply only to those who choose the monastic life. It's not necessary for lay Buddhists.

It tells to go forth alone (as a monk) fleeing ignoble companions:

In the midst of companions
— when staying at home,
when going out wandering —
you are prey to requests.
Valuing the freedom
wander alone like a rhinoceros.

There is sporting (flirting) & love
in the midst of companions,
& abundant fondness for offspring.
(While) feeling disgust
at the prospect of parting
from those who'd be dear,
wander alone like a rhinoceros.

People follow & associate for a motive.
Friends without a motive these days are rare.
They're shrewd for their own ends, & impure.
Wander alone like a rhinoceros.

However, it praises the company of noble companions and if there are none, better go alone:

If you gain a mature companion,
a fellow traveler, right-living & wise,
overcoming all dangers
go with him, gratified, mindful.

If you don't gain a mature companion,
a fellow traveler, right-living & wise,
wander alone
like a king renouncing his kingdom,
like the elephant in the Matanga wilds,
his herd.

We praise companionship — yes!
Those on a par, or better,
should be chosen as friends.
If they're not to be found,
living faultlessly,
wander alone like a rhinoceros.

Sometimes the monks wandering alone in the forest become lonely, and they need to be reminded, as we see in SN 9.9 below.

On one occasion a certain monk, a Vajjian princeling, was dwelling near Vesali in a forest thicket. And on that occasion an all-night festival was being held in Vesali. The monk — lamenting as he heard the resounding din of wind music, string music, & gongs coming from Vesali, on that occasion recited this verse:

I live in the wilderness
all alone
like a log cast away in the forest.
On a night like this,
who could there be
more miserable than me?

Then the devata inhabiting the forest thicket, feeling sympathy for the monk, desiring his benefit, desiring to bring him to his senses, approached him and addressed him with this verse:

As you live in the wilderness all alone
like a log cast away in the forest,
many are those who envy you,
as hell-beings do,
those headed for heaven.

The monk, chastened by the devata, came to his senses.

Here on the notion of solitude in Pali texts;

All the directions are obscure, The teachings are not clear to me; With our benevolent friend gone, It seems as if all is darkness.

1035. For one whose friend has passed away, One whose teacher is gone for good, There is no friend that can compare With mindfulness of the body.

1036. The old ones have all passed away; I do not fit in with the new. And so today I muse alone Like a bird who has gone to roost. (Ven Ananda's gatha)

Ananda, a monk does not shine if he delights in company, enjoys company, is committed to delighting in company; if he delights in a group, enjoys a group, rejoices in a group. Indeed, Ananda, it is impossible that a monk who delights in company, enjoys company, is committed to delighting in company; who delights in a group, enjoys a group, rejoices in a group, will obtain at will — without difficulty, without trouble — the pleasure of renunciation, the pleasure of seclusion, the pleasure of peace, the pleasure of self-awakening. But it is possible that a monk who lives alone, withdrawn from the group, can expect to obtain at will — without difficulty, without trouble — the pleasure of renunciation, the pleasure of seclusion, the pleasure of peace, the pleasure of self-awakening. (https://zugangzureinsicht.org/html/tipitaka/mn/mn.122.than_en.html)

...the Blessed One said to him, "Is it true, Elder, that you live alone and extol the virtues of living alone?"

"Yes, lord."

"But how do you live alone and extol the virtues of living alone?"

"Lord, alone I enter the village for alms, alone I return, alone I sit withdrawn [in meditation], alone I do walking meditation. That is how I live alone and extol the virtues of living alone."

"There is that way of living alone, Elder. I don't say that there isn't. Still, listen well to how your living alone is perfected in its details, and pay close attention. I will speak."

"As you say, lord," Ven. Elder responded.

The Blessed One said: "And how is living alone perfected in its details? There is the case where whatever is past is abandoned, whatever is future is relinquished, and any passion & desire with regard to states of being attained in the present is well subdued.[2] That is how living alone is perfected in its details." https://www.accesstoinsight.org/tipitaka/sn/sn21/sn21.010.than.html#fn-1

These are relevant to the question and afaik have not been recited in the other answers. So i post to complement the other good answers which are based on texts known as true.

Loneliness is not being alone. You have yourself, and that is the best company in existence.

You just have to become familiar with yourself, your mind without following anything, without grasping at anything, in a state of pure non-action. Leave everything as it is, and allow your true self to arise with splendor.