35W for 12 hours is 420 W*hr. One Li-ion 18650 cell (same as in many laptops) has capacity of about 10 W*hr. So you will need about 50 cells to keep your fan running for 12 hours. Each cell is about 45g, so the whole battery will weight about 2.2 kg, or about 5 lbs.
Now about solar. As I understand, you likely want to recharge your battery over one day, ready for another 12 hours. To recharge a 420 Wh battery over a daylight time, say 6 hours, your will need at least a 100-W solar panel, something like used for RV or boat, example on eBay. The thing will have 40" x 26" x 1.2" in size, and weight of 19 lbs (8.5 kg). Most likely you will need a bigger panel, since this one makes only 300 Wh over a sunny day, so you might need to double the panels.
ADDITION: I forgot to mention that in order to"create" a self-sustainable solar power system, you will also need:
(a) solar controller/stabilizer;
(b) balanced battery charger (to charge all cells evenly);
(c) power management electronics, to balance input between the charger, battery, and outlets;
(d) durable enclosure with proper connectors/outlets;
(e) mounting harware for solar panel, to follow the Sun, at least manually.
The project will take likely a full year for a team of several engineers.
Alternatively you can shell out some sizable money ($500 - $1000) and get a professionally built product, like the 400 Wh "Goal Zero Yeti solar station".