Can I set a standard object field as required using a custom layout?

No. Standard objects such as Leads, Opportunities, Competitors, and so on already have their associated field properties set, such as default, required, updatable, and read-only. Hence, a field for standard objects can't be set as required if using a custom layout.

You can use Page Composer to display a field with the required symbol (*) but it won't act as a functional required field. The required * symbol will be visible on the field and will allow manual data entry but without any functional implications.