Electronic – How to Choose a Battery

Your 75W fan required 220 Volts, it will not operate at typical battery voltages such as 12V.

A 75W fan operating (indirectly) from a 12V supply would draw 6.25 A not 0.34 A.

Even with a theoretical 100% efficient invertor, the current on the 12V side is greater than the current on the 220V side to deliver the same power across the system from battery to Fan.

Note that the reason that power is delivered through the grid at high voltage is to keep currents low. losses and cable sizes are proportional to (a function of) current. High V are used to keep currents low.

You convert 12V DC into 220 V AC using your inverter. Depending on capacity and utilisation the efficiency of an inverter might be the 80% you estimate or it may be worse - 50% Ref So your 6.25A may become 13A drawn from the 12V battery.

Therefore your 300 Ah battery-set might power a 75W fan for less than 24 hours.

Update: for 2 x nominal 12V SLA battery, the following diagram may clarify what is happening.

^{simulate this circuit – Schematic created using CircuitLab}

Let's clear up some terminology first. An Amp is not a discrete quantity of energy. It is actually a quantity of electrons per time. Therefore, you don't say "Amps per hour". I like to use the analogy that Amps of current is like speed in your car. If you drove your car for an hour at 60mph (pardon my English unit assumption) and someone asked you about your drive, you wouldn't say "I drove 60mph per hour". That would be silly. Instead you would say "I drove 60mph for an hour."

It's the same with current. A battery doesn't supply 5 Amps per hour, it provides 5 Amps for an hour. Furthermore, a battery that has supplied 5 Amps for an hour has provided 5 Amp-hours (5Ah) of charge. An Amp-hour is an actual discrete quantity. It represents the capacity of the battery.

So you need a battery that can provide 5 Amps for 2 hours at 12V. That's 10 Amp-hours total. More specifically, since it's 12V: $$5A*2hours*12V = 120Wh$$
Of course, you would never want to buy a battery that can only provide exactly the capacity you need. You want some capacity in reserve. So let's double it and say you probably want a 240Wh battery. That way, you'll only use 50% of the capacity in one session.

The Li-ion battery you mentioned in your question has a 9800mAh capacity at 12V. That works out to 117.6Wh. Almost enough to last for two hours with a 5A current draw. As long as you understand what you're doing with a Li-ion battery, that should work, but I would recommend putting two in parallel to double the capacity. When you put two batteries in parallel, their voltage stays the same (12V) but the total capacity doubles (9800mAh * 2 = 19.6Ah).

EDIT: Looking at the website for that Li-ion battery, it is not clear what its continuous current capability is. It may not be able to supply 5A of current continuously without overheating. Without a datasheet or more technical specs, it's probably not safe to use.

One other note, batteries are not perfect energy sources. As you discharge them, their voltage lowers, their internal resistance increases, and their ability to supply current decreases. The more current you draw at a time, the more these negative effects occur. What all that means is a 10Ah battery may not actually supply a full 10Ah before the battery is flat. This is another reason to get a battery that has a larger capacity than the math dictates.