What does the Millennium Falcon use for fuel (and how big is the fuel tank)?
According to the (canon) Force Awakens: Incredible Cross-Sections, the Falcon uses
liquid metal fuel
to power its sublight engines. There's no indication which metal.
You can see the fuel tank on this earlier (and non-canon but still excellent) plan from Star Wars: Incredible Cross-Sections. Assuming it's hollow, the tank depicted appears to be around 3-4000 litres in volume.
The Hyperdrive engine is powered by ejecting
The tanks are sufficiently large for them to cross half of the galaxy without refuelling.
TLDR: liquid metal fuel
As suggested by @JohnP, From the Wookipedia, YT-1300 light_freighter it is stated that the YT-1300 (the model of Millennium Falcon) use fuel cells powered by liquid metal fuel.
Another canon source is the Star Wars Millennium Falcon YT-1300 A 3-D Owner's Guide written by Ryder Windham himself :