Pico-itx motherboard Powered With Battery
Your load requires 12 V at 2.5 A to power it.  To power it for 4 hours therefore takes 10 Ah, at 12 V of course.  Your math is therefore correct.  However, that does not mean you should get a "10 Ah" battery.  Depending on the exact battery chemistry, running a battery totally down can cause permanent damage.  This is true of typical "car" 12 V lead-acid batteries, for example.

Fortunately, 12 V car batteries start a higher than 10 Ah.  A 20 or 30 Ah car is a choice for running your device for 4 hours.  There are also "deep cycle" batteries intended to be drained closer to empty, but of course these usually cost more for the same capacity.

Your question about "huge" and "heavy" can't be answered since you haven't defined those terms.  We do engineering here, not hand waving.  Magic is clearly off topic, and so are product recommendations.