Exceptions exist for private clubs according to https://en.wikipedia.org/wiki/Civil_Rights_Act_of_1964 but with regard to businesses getting to decide which customers they serve, it looks kinda like segregation on the basis of race (or gender, religion, or nationality) was outlawed.
Or the military has someone who is disabled, or literally any service or group that has specific physical needs such as Amazon Warehouses or UPS/USPS, race isn't the only segregation method.