United’s 24-hour flexible booking policy
and Ticket refund policies - Booking service fees

Note: United refers to this as "United’s 24-hour flexible booking policy", not just cancellations. UA includes routing changes, pricing changes, fare class changes, ..... as well as simple cancellations. But the timeframe is based on the original purchases and not reset by a change. Only a cancellation and then separate rebooking would achieve that.

Reservation booking service fees and close-in booking fees for award tickets are refundable only if you cancel your reservation within 24 hours of purchase. These sometimes may take an extra request.

Starting 2016Q4 ticket receipts state The website link does now mention the 7 day restriction (new 2017Q2) and the CoCs has contained the 7 day restriction for awhile. This is allowed by the DoT rules. However, no reports yet of UA enforcing the 7 day restriction.

DOT 24 refund rule, announcement
76 Fed. Reg. 23110, 23166, Apr. 25, 2011

Further DOT Guidance on 24 hour rule
Note the present UA implementation in practice is more generous -- no 7 day limitation enforcement as of yet.
UA's 24 hour rule was in place long before the DOT requirement .

Note -- a changed ticket does not get a new 24-hour free cancel period -- that benefit only applies to the original purchase.

Does the 24 hour rule apply to award tickets?
Yes

Does the 24 hour rule apply to close in bookings?
Yes but cancellation must be done before scheduled departure.
While the policy revision in 2018 add the DOT allowed 7-day exemption, UA has generally allowed this and not enforced the 7-day exemption. There has been just one FT reported situation where the 7-day was enforced and it had other factors.

In the case of credit card purchases, on a cancellation, the credit charge may be left pending and never finalized. So in the end it is as if it never happen.

Basic Economy tickets -- are they projected by the 24 hour rule?
Yes, UA is allowing cancellation of BE tickets in the first 24 hour (consistent with DOT requirements. But as UA allows changes also of other fares in the first 24 hours, it will not allow that of BE . The distinction is minor but

Book a reservation with ETC but cancelled with 24 hours, will the ETC be refunded?
Yes it should but UA can be rather slow on this and occasionally will drop the ball. You may need to followup. Note the refunded ETC will have the same expiration as the original ETC. (So, not a way to extend the ETC )