Sizing a battery for given solar panels?

Those are 12V panels
What vltage battery bank do you plan to run.
If 12v put them in parallel with a PWM controller or in series with an MPPT controller
If 24V series in both types of controller.

You may not be wrong, but you are not right either.
The answer will depend on the kind of Charge Controller (CC) you put between the panels and the batteries, among other things.
You cannot connect panels directly to the batteries without the near certainty of frying the batteries and/or the panels.

If the battery bank is 24 volts, then you certainly have to put the panels in series.
If the battery bank is 12 volts (suitable for a small system only), then you will put the panels in parallel if you use a PWM CC and in series if you use an MPPT CC.

With 270 watts of panel, you should, according to rule of thumb, have a roughly 270 amp hour battery bank at 12 volts or a 135AH battery bank at 24 volts.

But the first step in designing an solar PV system with batteries (well the second step after figuring out whether you really want to go down this route at all) is to figure out what your power loads will be during the course of a typical day.
That allows you to size your battery bank and inverter.
Then you can size your panels and CC.
Then and only then should you actually buy any of the components.

And, by the way, welcome to Solar Panel Talk!

I calculated the average daily solar radiation in my area which is 4.29 kWh/m^2/day
As for my typical daily power load, I would like to use the battery to power any appliance, in other words I would like to use it to its full potential. So I don't mind powering anything from light bulb to a refrigerator.

I like the idea of connecting them in series to a 24 Volt battery though

Also, thanks for replying I appreciate all your help.

You cannot use Average Radiation for a battery system. It has to be worse case winter month. Otherwise you go dark in winter.

The worst solar radiation is 2.13 on average, and I'm also assuming 5 days with no sun.

Given what you have
1- refrigerator is out of the question. You cannot supply even one day of running it with the panels you have.
2- At 5 days without sun the battery bank would be huge or you will over discharge the batteries and destroy them quickly.
3- During the worst month given what you have for panels you can run with a PWM controller a total of 300 watt hours a day or 360 watt hours a day.
3- for 5 days of autonomy and keeping your batteries up above 50% DOD would require a battery bank of at least 150 AH @24V or 300AH at 12V
4- Your panels would need to to double to provide enough charge amps to properly maintain the batteries.

Sam you are heading for big trouble. What you plan to do is grossly undersized and nothing more than a down payment for what is needed. You need to completely rethink this.

Ya

My job was to build a prototype using the two panels, specifically the mechanical and the designing of the prototype was my job (i.e. frame, mounting, tracker, durability)
but come the electrical part I have no idea what I'm doing.

To be honest, I'm not really looking for suggestions, I'm looking for straight answers. I have the prototype (with the panels shown above), what should I power with them, what battery/inverter/wires should I use?

I don't mind making some assumptions!! As long as this prototype powers something i.e. proof that it works, I'll be a happy man.

Thank you all for being helpful and patient

If this is a school project say so.

Lights. Lots of yard lights. Cheap, and if it fails, no food is lost. Otherwise, you are drinking warm beer.

There is no straight answers as you have not really given enough information. First thing you left out is a charge controller which you must have. So making the assumptions you will use a inexpensive PWM controller, using 270 watts of battery panels wired in parallel, and with a 2.13 Sun Hour Winter day here is what you get.

12 volt 120 AH battery
20 amp PWM controller

This system can only generate around 300 watt hours of usable energy per day or around 3 to 4 cents worth of electricity. Put another way run a 60 watt light bulb for 5 hours.

Or a fairly bright 20W LED throughout a 15hour winter night...