I've used SUMIF for simple calculations such as:

SUMIF($F$32:$F$39,"Contracted",$H$32:$H$39)

on a single sheet.

How would I work that using INDIRECT to perform that same function for
multiple sheets?

I can't seem to make it work using a similar format to:

SUM(INDIRECT($B6&"!$H$17:$H$20") where B6 is the worksheet name I want to
reference.

In my example I am trying to sum the dollar amounts in column H for those
services that are labeled <Contracted in column F across multiple worksheets
(ie. for each worksheet, I want one total dollar amount).

Is it SUMIF plus INDIRECT or is there another way to go about it?