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United Airlines Basic Economy is really, really basic. In exchange for price, you agree to give up all sorts of options, including the carry-on baggage allowance. Nothing can be put in the overhead and not only must it fit under the seat, but the dimensions are smaller than many laptops, more the size of notebooks.
How strictly is the policy enforced? Ask United Airlines and it will quote the fare policy:
If you're traveling on a Basic Economy ticket and bring a full-sized carry-on bag to the gate, your bag will need to checked and you'll be required to pay the applicable checked bag fee plus a $25 gate handling charge (payment by credit card only).
Reports by experienced travelers and a look at consumer feedback (the links follow) suggest it is being done routinely, that United added a new Economy baggage sizer, and that, since seats assignment happens at the depature gate, agents do check. United Airlines has a reputation for being unbending about its rules.