Is Buddhism in conflict with manliness?

OP: To be man we have to compete, obtain resources and protect our families. Could it be all achieved reasonably and with compassion? Could a man following Dhamma be a real, good man?

What you describe here is basically known as a householder or the head of a household in old times, and maybe even today.

The Sigalovada Sutta has plenty of good advice:

The wise endowed with virtue
Shine forth like a burning fire,
Gathering wealth as bees do honey
And heaping it up like an ant hill.

Once wealth is accumulated,
Family and household life may follow.
By dividing wealth into four parts,
True friendships are bound;

One part should be enjoyed;
Two parts invested in business;
And the fourth set aside
Against future misfortunes."

"In five ways should a wife as the western direction be respected by a husband: by honoring, not disrespecting, being faithful, sharing authority, and by giving gifts.

"And, the wife so respected reciprocates with compassion in five ways: by being well-organized, being kindly disposed to the in-laws and household workers, being faithful, looking after the household goods, and being skillful and diligent in all duties.

From the Dighajanu Sutta:

"And what does it mean to maintain one's livelihood in tune? There is the case where a lay person, knowing the income and outflow of his wealth, maintains a livelihood in tune, neither a spendthrift nor a penny-pincher, [thinking], 'Thus will my income exceed my outflow, and my outflow will not exceed my income.' Just as when a weigher or his apprentice, when holding the scales, knows, 'It has tipped down so much or has tipped up so much,' in the same way, the lay person, knowing the income and outflow of his wealth, maintains a livelihood in tune, neither a spendthrift nor a penny-pincher, [thinking], 'Thus will my income exceed my outflow, and my outflow will not exceed my income.' If a lay person has a small income but maintains a grand livelihood, it will be rumored of him, 'This clansman devours his wealth like a fruit-tree eater.' If a lay person has a large income but maintains a miserable livelihood, it will be rumored of him, 'This clansman will die of starvation.' But when a lay person, knowing the income and outflow of his wealth, maintains a livelihood in tune, neither a spendthrift nor a penny-pincher, [thinking], 'Thus will my income exceed my outflow, and my outflow will not exceed my income,' this is called maintaining one's livelihood in tune.

From the Anana Sutta:

"And what is the bliss of having? There is the case where the son of a good family has wealth earned through his efforts & enterprise, amassed through the strength of his arm, and piled up through the sweat of his brow, righteous wealth righteously gained. When he thinks, 'I have wealth earned through my efforts & enterprise, amassed through the strength of my arm, and piled up through the sweat of my brow, righteous wealth righteously gained,' he experiences bliss, he experiences joy. This is called the bliss of having.

"And what is the bliss of [making use of] wealth? There is the case where the son of a good family, using the wealth earned through his efforts & enterprise, amassed through the strength of his arm, and piled up through the sweat of his brow, righteous wealth righteously gained, partakes of his wealth and makes merit. When he thinks, 'Using the wealth earned through my efforts & enterprise, amassed through the strength of my arm, and piled up through the sweat of my brow, righteous wealth righteously gained, I partake of wealth and make merit,' he experiences bliss, he experiences joy. This is called the bliss of [making use of] wealth.

"And what is the bliss of debtlessness? There is the case where the son of a good family owes no debt, great or small, to anyone at all. When he thinks, 'I owe no debt, great or small, to anyone at all,' he experiences bliss, he experiences joy. This is called the bliss of debtlessness.

From the Adiya Sutta:

'My wealth has been enjoyed,
my dependents supported,
protected from calamities by me.

I have given supreme offerings
& performed the five oblations.

I have provided for the virtuous,
the restrained,
followers of the holy life.

For whatever aim a wise householder
would desire wealth,
that aim I have attained.
I have done what will not
lead to future distress.'

When this is recollected by a mortal,
a person established in the Dhamma of the Noble Ones,
he is praised in this life and,
after death, rejoices in heaven.

By practising the five precepts and other things, the householder can achieve stream entry. Please read the Gihi Sutta for more info on this.