It would be very difficult to get anything like an accurate measurement without using calipers. They don't need to be anything fancy; these 6 In. Utility Caliper would be perfectly adequate.
buy some calipers. everybody needs calipers. other than that, one of those old-timey external spring calipers and a ruler will get you close but they cost more than a pair of econo calipers at the bargain bin of the hardware store or those $2.39 plastic vernier calipers posted above. at the end of the day you will see that you need calipers.
Get some cardboard. Make an L. Cut another piece that's an I. Place the L under the neck. Place the I above, touching the fretboard. Stick it there. Mark the centre point on both pieces. Remove the device and measure between the two centre points.
I find it amusing that when the question is how to measure without calipers, the answer is get some calipers.
Awesome suggestion. You've helped me a few times with setup questions, too. I really appreciate it. This is the kind of help that makes Talkbass so valuable.
Ha-Ha, in the P&E subforum folks ask what pickups they can get for $65 USD & folks respond with stuff like Frailins!
This is all quite amusing....Richard's elegant technique with the cardboard is actually the pure form of a caliper. An adjustable or temporary mechanism which is used to reach the two measuring points and be locked down. Then it's gently removed, and a ruler or other device is used to measure the gap. That's what a caliper is. It's used to transfer a distance away, to where it can be measured. These days, these things we call Calipers, are calipers with the ruler built in. Either a dial gauge or a digital readout. You adjust the jaws to contact the workpiece, and it internally transfers that distance to the ruler, and reads out a number. Modern technology, in use for about 150 years.
Context is important here. If the OP is measuring ballpark neck thickness simply to satisfy curiosity and no decisions of consequence are based on the measurement, then accuracy and precision aren't critical...and who doesn't love playing with scissors? But if, for example, the OP is about to order a new neck and wants to base that purchase on the size of an existing neck, that's another story and that story involves calipers.
You generally already have a caliper, if you're into measuring stuff, too... But yeah, needs context. Why is it so important to be able to measure, but exclude the most useful tool for it?
Support the neck on a table/block. Put a spirit level on the fretboard and check it. If it's all level, measure from the table/block surface to the bottom surface of the level.
Wrap a string around the neck. Mark the string precisely where the ends would meet each other, then measure the string. Probably some math involved once you have that. the other method involves removing the neck and immersing it in a bathtub that you have marked the water line with a sharpie...