Installing pull up bars

I am thinking about installing pull up bars in my lawn. For this purpose, my plan was to use 4x4 pressure treated posts (about 4' apart) and then put a bar between them at abt 7' ht. However, I have several questions:

How deep in the ground should the posts be.

Do I need concrete to hold them, or will the dirt I dig up/gravel be sufficient to keep them stable.
Another thought I had was to use 3 posts in a right angle triangle shape (each arm at right angles to be abt 4') or to use 4 posts to make it a 4' square, to make the structure more stable. Is that preferable, and if so which one (triangle or square)? I want to use min # of posts so as to use min amt of area in the lawn.

What is the best way to tie the bar to the posts?
Lawn is regularly watered. Is that a problem? Any precaautions I need to take?

Thanks for your help in advance.

AK asks:

I put in something similar for my grandchildren a bunch of years ago. Drilled the 4x4s about 1/16" over-sized for the threaded pipe used as a bar (1-1/2" OD). Then planted the bars about 40" apart, with a 60 lb. bag of ready-mix around the base of each one...after I leveled the bar. Holes were 24" deep, based on my estimates of kids' weights increasing, and frost heave depth--18" max here. To be honest, I think 18" overall would have been overkill once the concrete was solid and frost heave isn't going to do much on something that small.

Pipe was held in place with end caps.

I didn't mess with triangles. The lawn was watered then as now by rainfall of which central VA usually has a sufficiency that keeps the damned grass about

1-1/2" longer than is neat most of the time. Charlie Self "Inanimate objects are classified scientifically into three major categories - those that don't work, those that break down and those that get lost." Russell Baker