How long does Gnarled Sage's +0/+2 stick around for?

Gnarled Sage gets +0/+2 when you've drawn 2 or more cards this turn. It doesn't say “until end of turn” though, so will the +0/+2 stay or not after the end of the turn?

It lasts until the end of the turn and doesn't stay longer.

Gnarled Sage's ability is a static ability:

As long as you've drawn two or more cards this turn, Gnarled Sage gets +0/+2 and has vigilance.

It's simply checking if this condition is true. If it is true, the other thing is also true. We calculate this moment to moment whenever we need to know it.

As long as this is a turn you've drawn two or more cards on, it gets +0/+2 and vigilance. As long as this condition is not true—such as at the beginning of a new turn—it does not get these features.

This is not a triggered ability (“when you draw a second card each turn, CARDNAME gets +0/+2 ...”). That would apply a continuous effect, and the continuous effect would need to have an expiration.

But we're not doing that: we're just evaluating a condition in a static ability to see what's currently true (+0/+2 or not).

You're thinking about Gnarled Sage's ability as if it were a triggered ability that applies a continous effect. It's not. It's a static, conditional, ability.

The ability is always functional - it's just that it has no effect unless you've drawn two or more cards that turn. When the turn ends, the ability doesn't go away, but it's no longer true that you've drawn two or more cards, so Gnarled Sage stops getting +0/+2.

The buff to Gnarled Sage only lasts from when you've drawn your second card until the end of the current turn, even though it is worded differently. For most practical purposes, it's the same as an "until end of turn" triggered ability.

Most abilities on permanents that do something "until end of turn" are triggered abilities, meaning the ability waits for a trigger event and then does something, such as Steelgaze Griffin. By contrast, Gnarled Sage's ability is a static ability that is functional as soon as and as long as its enabling condition ("As long as you've drawn two or more cards this turn") is true.

However, the condition makes sure that it only activates the buff for the current turn, which comes out to the same in most situations as a hypothetical triggered ability that gives the same buff "until end of turn".

There are a few mechanical differences between the ability being worded as a static ability rather than a triggered ability, but as far as your question about duration is concerned, there is no practical difference. Whenever a turn ends and a new one starts, you will not have drawn 2 cards during the current turn, and the buff won't be active.