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The only measurement that matters regarding nut slot depth is how much clearance you have between the top of the first fret and the bottom of each string.
More exactly, each slot should be filed down until you have between .020" and .022" of clearance between the bottom of each string and the top of the first fret.
So, are your strings fretting out on the first fret when you pluck them open? If not, then they aren't too low. However, they may still be too high. You'll need some feeler gauges to really find out.
And your guitar didn’t implode? That can’t be right.
Joking aside, that’s awesome you did that. That’s how I started too. I’d recommend getting something like this. It’s pretty inexpensive and will be what you use the most for the very basic setup related stuff, like adjusting the action and the neck relief (truss rod).
Totally agreed - the cost of a basic setup at most music stores is around $50 or so, but you can purchase all of the tools needed for less than $20. All you need is -
Most manufacturers even publish a step by step guides on how to set up their instruments to factory specifications. For example, here's a link to Fender's bass setup guide.
Every couple of months, I get up early on a Saturday morning, get some coffee going, and do a full setup on my P Bass, Stratocaster, and McCarty 594. Those are some seriously relaxing mornings.
One tip I can recommend when measuring the relief, is to do it in playing position, so don't measure it while it's laying flat on a table with the neck cradled or anything, measure it while it's on your leg. Here's the official Fender bass setup guide:
https://support.fender.com/hc/en-us/articles/214343843-How-do-I-set-up-my-bass-guitar-properly-
Capo, automotive feeler gauges, and if you plan to also adjust action (bridge saddle height, which I recommend setting after you get the truss rod set) get a cheap string action ruler for that too. You can get both the ruler and feeler gauges for $10 on amazon: