It is, in fact, possible. [e]at the blood...obviously, as implied, use the [e] key to bring up the eat/drink function. You will see an inventory-like interface with everything in your inventory (attempting to eat or drink an inedible item such as a knife or a cloak will prompt your character to lick said item) including blood/water/ichor coatings on your body parts/weapons/armor/whatever. Drinking a tiny spatter of blood on your dagger will quench your thirst equally as much as drinking from a flowing river. Some examples of what you may find in the [e]at interface:
a. Drink water [Here]
b. Drink stray cat blood [NW]
c. Drink water covering [Giant armadillo leather sock]
d. Drink blood spatter [iron longsword]
e. Drink giant grasshopper ichor [E]
Scroll through and you can also drink water from waterskins/backpacks/jugs/pots/whatever other container you've filled with water. To fill a container, {I}nteract (note the uppercase letter) with the container and there should be an interface that says something along the lines of:
Fill waterskin with water/blood/ichor......NW
Fill waterskin with water/blood/ichor......Here
Etc. You can infinitely fill containers from one puddle of liquid (I believe) but they obviously have a limit on how full they will fill. {I}nteract again to empty a container of a liquid.
The type of blood drinking you are thinking of (the vampiric kind, I mean) uses a slightly different mechanic than what you require...when your character is a vampire (to become one, simply drink the blood of any vampire that has bled recently using the same technique discussed above [this is rather buggy and doesn't always work, sometimes requiring the player to find a different vampire and drinking his or her blood {if you are unsure whether or not you are a vampire, find a place to sleep safely and sleep for eight or more hours...if you have not accumulated any thirst or hunger, congratulations, you're a vampire}]) you will accumulate no thirst, hunger, or drowsiness (sleep), however after a more extended period of time your character will become "Thirsty." In order to quench your vampiric thirst, find any unconscious creature (either beat one into unconsciousness by causing extreme pain [hint: only use blunt attacks and shatter bones to avoid blood loss] or find a sleeping creature) and use the [e]at command. At the top of the interface will be the option:
a. Suck elephant/elf/goblin/whatever blood
Which will often quench your thirst. Smaller creatures quench less and tend to be "completely drained of blood" without sating your bloodlust, so it's recommended you find a medium to large sized creature to feast on. Companions will attack you if you drink from sleeping innocents but do not care if you drink from an unconscious enemy/wild animal.
Also, a couple more tips. 1. Wells are f***ing impossible to find, so if you're looking for an unfrozen water/liquid source, use [g] to create a campfire and {I}nteract with your waterskin to heat the liquid and melt it. 2. Killing stray cats or any other domestic animal will condemn you in the eyes of the locals for whatever reason, so it's recommended that you avoid slaying them and search for another source of water. 3. [BONUS]
If you're in the wild and starving, but just happen to come across an easily-defeated wild animal, [r]emove a cutting implement from your bag (knife, dagger, spear, arrow, sword...anything with even a slight edge), position yourself directly over the animal, and hit {x} to open the action interface, select butcher, select the corpse in question, then select a cutting tool and you now have a nice big pile of meat and prepared organs. The body doesn't have to be whole either, you can butcher a mutilated corpse, then walk over to a severed limb and butcher that too. This is, I've found, the easiest way to remain well-fed while adventuring. You can even butcher corpses you're carrying in your bag using the same method (be careful with unbutchered meat though, necromancers can reanimate them while they're still on you're person, resulting in an unwelcome sneak-attack and lots of !!FUN!!).
I hope this helps
Edit: Typo