i am thinking of making my own handheld out of a intel NUC and also a mxm gpu attached, one problem i had is that the apparent tdp altogether is like 120w. If I want a 4 hour (at least) bettery life, how big of a battery would i need?
That's not even remotely doable in a handheld form factor - or in a portable form factor at all. Remember, most gaming laptops severely limit their power draws when running off battery, and still last for an hour or less. For any reasonable battery life while gaming you need APU-level power draws - 15-35W at most.
A reasonable reference, the Asus Flow Z13, with a 56Wh 4-cell battery, a 12900H (operating in a low power mode when running off battery, limited to 30W) and a 35W (+5W boost) RTX 3050 Ti
lasted 47 minutes playing The Witcher 3 in Notebookcheck's review. That tablet barely exceeds 100W total (including the display, I/O, motherboard, RAM, and everything else in addition to the CPU and GPU) during stress testing, and consumes ~70W in their Witcher tests. It's pretty easy math, really: you'd need a 120W*4h=480Wh battery to sustain a 120W load for four hours. That's a
huge battery.
Here's an example of a 12V 480Wh LiFePo battery kit, and while it's clearly not optimized for low weight but rather durability, it weighs 6.4kg/14.1lbs. You can drop that mass significantly with a higher voltage battery pack, with 48V ones ranging from ~1.7-2.5kg from what I've seen, but then you need to contend with conversion losses getting the voltage to a usable range, plus the space for a high current buck converter (and its heat output). And, of course, 2kg+the weight of your hardware isn't even remotely doable for a handheld - even things like the 825g OneXPlayer gets complaints for being too heavy.