Is it possible to believe that there are no absolute truths?

You can believe it, as you can believe pretty much any false statement. But yes, saying there are no absolute truths is self-refuting, absolutely.

You're under the misapprehension that statements are self referential. This is a version of what is known as the liar paradox. Consider the sentence "This sentence is false". 'This' is what philosophers of language call an indexical phrase. If the 'this' is allowed to index onto itself, then the sentence is only true if it is false, and only false if it is true, hence the paradox. However, the paradox supplies its own solution. Allowing an indexical to index to itself is an invalid recursive function, and so as a practical matter, nobody does it. "This statement is false" can only meaningfully be said of a sentence other than itself, and then there's no problem with its meaning. Same as "There are no absolute truths". It's a quantifying indexical phrase that cannot meaningfully include itself. This is related to Russell's paradox.

In any case, though the sentence is not self defeating as many imagine it to be, I think it's straightforwardly false. Do you mean to suggest that there are no true statements? How about trivial statements of identity, like 'A is A'? Also, do you have a Y!A account? If yes, is it TRUE that you have a Y!A account? Things like this may not be spiritually or intellectually moving truths, but truths they certainly are nonetheless.

If there are no absolute truths, that means all knowledge is completely subjective/relative. In that case, absolute truths exist only in the eye of the observer. If the absolute truth exists only in the eyes of the observer, then they are not absolute truths at all; but only truth that the observer believes is absolutely true.

Talk to any Buddhist/Hindu about this!

Try arguing this with the guy who's standing on the street corner, talking to himself while scratching his head. You say potato, he says intergalactic grenade. You tell him the sky is blue and he tells you left hand turns went out with purfomasiclacticbypanorism (don't look it up, I made it up).

Essentially, yes the statement does contradict itself. Such is the source of many "great" debates, which in truth are nothing more than half-baked attempts to appear intelligent.

The principle of the statement is foundationally sound. What do you believe is true? Up is up? Try that in the vacuum of space. Oh, you think you're really here, don't you? Well "mathematically" I can disprove that because on the atomic level you are 99.9% empty space. I just gave you what you believe is a sound argument - mathematics - but I posit you are putting your faith in something designed and instituted by the fallible. If you think mathematics is perfect (and if it isn't, it can't be trusted to be correct), try solving one of those unsolvables, or use it to PROVE the THEORY of General Relativity.

Humanity has only one thing on which to base its "truth" - itself. Is that really truth?

Statements of facts, like truth being relative, can never be absolute. You can think that, but thought isn't the truth, it has to be realized when you stop thought and the division between you and what is is gone. You can also say Truth is always new and it's a continual realization, but that's done with consciousness and not thought. Thought can never know truth.

Yeah. Watch "There are possibly no absolute truths." This is not an absolute truth it simply takes anything that would be called 100% true and dials it down to 99.9999999% or whatever.

It is possible to believe that there are PROBABLY no absolute truths. But not ABSOLUTELY.

As for me, all I know is that I know nothing.

- Socrates

Like Socrates, I don't know any absolute truths, and don't expect to discover any in my lifetime.