Is there a convenient way of getting rid of water?
I tried draining it in lava but that didn't work, some impenetrable stone was created...

Water + Lava = Obsidian.

You will need at least a Nightmare/Deathbringer Pickaxe to mine it. Or you can use bombs/dynamite to blow it up. You will get the materials to make a Nightmare Pickaxe after defeating the Eater of Worlds (available if you have Corruption in your world) and the Deathbringer Pickaxe after defeating the Brain of Cthulhu (available if you have Crimson in your world). You're almost guaranteed to need one of those two pickaxes anyway since they're able to mine Hellstone, you'll need that and Obsidian to make Hellstone Bars which is the main component of the highest quality pre-Hardmode gear.

But if you just want to completely remove water, mine all the way down. Water will start evaporating if it flows in the Underworld.

You can also use pumps to move water, but as WingedKagouti pointed out, water will evaporate in the underworld.

Honey also works.

Start evaporating?
That sounds good!
When and how exactly does that happen?

The bottom of the map? Like, all the way down, where the map ends? Doesn't that take ages? Can you get rid of lava the same way?

Again, not to be a broken record but using honey allows you to turn puddles of water into puddles of regeneration not to mention all the cool stuff you can make with it.

Yes. Yes. Yes. No.
Some more details:
Yes, you need to go to the bottom of the map and it will take a long time (unless you are very far in the game, but you say that you can't break obsidian so I assume that you aren't). You can also just try to find some caves and dig further down. Otherwise, you can use bombs to dig. If you don't want to do this, you can always just dig a deep hole and let the water flow down there and then remove it later. As far as I know, you can only get rid of lava by pouring water / honey on it.

I didn't even destroy Eye of Cthulhu until now... I once had a Gold Pickaxe, but dug myself into lava and was roasted and lost all my gear.
Yeah, stupid.
Many thanks for your help!

Dig a 2x2 hole right underneath the water, put in a chest, dump a pail of lava on the chest, then mine away one of the tiles to let the water pour in.

Simplest method, short of draining into a cavern, is to create a bunch of buckets out of iron/lead bars, fill them up, and dump the water somewhere else. This works for lava and honey as well. I care 20 buckets around with me at all times. Very handy when digging a hellevator.