About a DIY off grid solar power system for a small cabin for only $580.00?

I just discovered they sell 12 Volt, 35 Amp Hour Sealed Lead Acid Batteries at Harbor Freight for $69.00. If I were to get 4 that would be about $280. The only thing is that these batteries are only 35 amp hours. However, I think this rig would work. The reason is that I only plan to run a PC (Dell Inspiron 530), a... show more I just discovered they sell 12 Volt, 35 Amp Hour Sealed Lead Acid Batteries at Harbor Freight for $69.00. If I were to get 4 that would be about $280. The only thing is that these batteries are only 35 amp hours. However, I think this rig would work. The reason is that I only plan to run a PC (Dell Inspiron 530), a modem (Linksys Cable Gateway), and a tiny fridge. The specific fridge is the Magic Chef 3.6-Cubic Foot Refrigerator, which has an estimated yearly energy cost of $27.-.

For the sake of this thought experiment let us imagine that I will be running the computer, modem, and fridge 24/7.

The components I imagine would work:

250W solar panel (I found one for $180.00 locally).

Charge Controller: 30 Amp Solar Charge Controller
THUNDERBOLT MAGNUM SOLAR - Item#68738 (@$79.-).

Inverter: 750 Watt Continuous/1500 Watt Peak Power Inverter
Cen-Tech - Item#66817 (@$41.99).

Battery Bank:12 Volt, 35 Amp Hour Sealed Lead Acid Battery
THUNDERBOLT MAGNUM SOLAR - Item#68680 (4 @ $69.99 so about $280.-)

Total: $580.-

Ultimately, my issue would be the few hours where there is no light. If my batteries are 35Ah, then I should be able to use 17.5Ah before hurting the battery bank, right (I am new at this)? Would 17.5Ah hours be enough reserve capacity to carry me through the dark hours of the night (running those 3 items continuously)? Until the sun rises and recharges my battery bank and the whole process starts again? Or, should I get batteries with more Ah?
Update: Where I live the shortest day (on the winter solstice) is about 12 and 1/2 hours.

DO YOU SEE ANY OTHER ISSUES WITH MY PROPOSED SYSTEM?
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