How fast do I actually need to be able to reach my emergency money?

Fast enough to handle the emergency without borrowing, which usually means same-day for a small slice and one to two business days for the rest. Keep a small instant-access buffer, perhaps one to two thousand dollars, in checking or a linked savings account so a car tow or urgent copay never forces you to wait on a transfer. The larger portion can sit in a high-yield savings or money market account that settles in a day or two, which is fine for a job loss or a repair bill you will pay over a few weeks. Money you would need in literal minutes should not be in a CD, a brokerage, or anything with a settlement delay. Test your setup once so you know exactly how long a transfer takes before you actually need it.