How far is too far when it comes to drinking alcohol?

I'd advocate that it's at your own discretion. I don't think it can be set in stone what exact amount of alcohol intake would warrant drunkenness or tipsiness.

Different people, I've learned, react differently to different amounts of alcohol in their system. Some can control themselves better than others.

But as a general rule of thumb, it would be appropriate to go with Sotiris' advice in his answer. So long as you are clearheaded, and it doesn't stand in the way of your worship, then you should be fine.

The Bible have a lot of places where drinking wine is a natural thing, but often referenced in context of reduced self control.

Genesis 9:21
When he drank some of its wine, he became drunk and lay uncovered inside his tent.

Esther 1:10
On the seventh day, when King Xerxes was in high spirits from wine, he commanded the seven eunuchs who served him—Mehuman, Biztha, Harbona, Bigtha, Abagtha, Zethar and Karkas—

We also have to remember that wine was a natural choice for drink to food.

Ecclesiastes 9:6-8
6 Their love, their hate and their jealousy have long since vanished; never again will they have a part in anything that happens under the sun.
7 Go, eat your food with gladness, and drink your wine with a joyful heart, for God has already approved what you do. 8 Always be clothed in white, and always anoint your head with oil.

What is discouraged is not necessarily to NOT drink wine at all, but do not drink too much wine:

Proverbs 23:19-21
19 Listen, my son, and be wise, and set your heart on the right path:
20 Do not join those who drink too much wine or gorge themselves on meat,
21 for drunkards and gluttons become poor, and drowsiness clothes them in rags.

For people with high positions or meeting before God, there are often more strict rules:

Leviticus 10:8-10
8 Then the LORD said to Aaron, 9 “You and your sons are not to drink wine or other fermented drink whenever you go into the tent of meeting, or you will die. This is a lasting ordinance for the generations to come, 10 so that you can distinguish between the holy and the common, between the unclean and the clean,

Proverbs 31:4
It is not for kings, Lemuel — it is not for kings to drink wine, not for rulers to crave beer.

But at the end, it is important to respect others view on this, no matter your own opinion. The following verse applies more or less to all questions about food and drink for us Christians:

Romans 14:20-22
20 Do not destroy the work of God for the sake of food. All food is clean, but it is wrong for a person to eat anything that causes someone else to stumble. 21 It is better not to eat meat or drink wine or to do anything else that will cause your brother or sister to fall.
22 So whatever you believe about these things keep between yourself and God. Blessed is the one who does not condemn himself by what he approves.

Personally, I practice to never drink so much that I get drunk. If I am with people I know practice a total absense, I don't drink at all to show respect for their view.

Common English Bible (CEB) uses an interesting word when translating 1 Peter 5:8

Be clearheaded. Keep alert. Your accuser, the devil, is on the prowl like a roaring lion, seeking someone to devour.

This and other verses exhort us to be prudent even with alcohol.

Some more pertinent verses are the following:

1 Thessalonians 5:6-9

So then, let us not be like others, who are asleep, but let us be awake and sober. 7 For those who sleep, sleep at night, and those who get drunk, get drunk at night. 8 But since we belong to the day, let us be sober, putting on faith and love as a breastplate, and the hope of salvation as a helmet. 9 For God did not appoint us to suffer wrath but to receive salvation through our Lord Jesus Christ.

2 Timothy 4:5

But you, keep your head in all situations, endure hardship, do the work of an evangelist, discharge all the duties of your ministry.