If we consider the speed of light, denoted as "c," to be equal to the pure number 1 without any units, it implies using a system of natural units. In natural units, fundamental physical constants such as the speed of light, Planck's constant, and the gravitational constant are set to 1, simplifying mathematical expressions and removing the need for units.
In such a system, the units used to measure space and time are chosen to maintain consistency with the equation c = 1. Since the speed of light traditionally has units of meters per second (m/s), this implies that the units of space and time should be chosen so that their product yields a dimensionless quantity, i.e., a pure number.
One common choice in natural units is to use units where the speed of light, c, is set to 1, as you mentioned. In this system, both space and time are measured in terms of the same unit, and their combination is dimensionless. It essentially treats space and time as interchangeable and simplifies certain calculations in theoretical physics, particularly in the context of relativity and quantum field theory.
Therefore, if we consider c = 1, the units of space and time would be the same and chosen so that their combination yields a dimensionless quantity.