How can I run an unswitched receptacle from a three-way switch box?
How is this switch wired? Is it a 3-way switch or a simple on/off switch? (It's a 3-way switch if it is one of two switches that can toggle your lights on/off. It's a simple on/off switch if there are no other switches that control the same lights.)

If it's a 3-way switch, then its two positions toggle a "common" conductor among two "traveler" conductors. Usually white is wired to common (one side of the switch) and black and red will be different travelers.

In that 3-way configuration, the box containing the circuit does not have a neutral available. The neutral will be available at the light or at the other switch box. This means you cannot run a new outlet from the wires in this box - you need a neutral wire.

If it's a simple on/off switch with 3 black/white/red wires, then it's possible that you have a neutral (which will not be connected to the switch because all the switch needs to do is connect/disconnect a hot conductor). Typically that will mean your white neutral wire is unused (should be capped with a wire nut) and the switch will connect black to red when turned on. If that's the case, then you can run a new outlet by connecting to the white neutral wire and the black (always hot, not switched) wire. You could also run another switched fixture from this outlet using white and red (switched hot) wires.