The "observatories" each consist of two 6' diameter stainless steel vacuum "beam tunnels", each 2.5 miles long and about 6 ft. in diameter, at right angles to each other.
The difference in gravitational waves in the interferometer is expected to be several orders of magnitude smaller than a human hair.
How much is this costing the taxpayers?
Looking at the web site for the NSF, it appears that spending so far is about $400 million. That's from the NSF. It is virtually impossible to find out if other government agencies are sharing the cost.
Your tax dollars at work.