Yes, you are talking about DC signals, and a driving a device (relay) that are way too slow to react to any reflections or care about the rise and fall times of the signals.

When talking DC, then it is enough to just look at the DC current capabilities of the LS245 and the resistance of the cable. And the resistance for 2-3 meters is quite low. At higher speeds, you need to think about how fast a wave moves in the cable, since the cable length with decide the time until the reflection returns. And the impedance of the cable in relation to the termination will decide how large percentage of the signal that gets reflected on each side, which decides how many times the reflections may ping-pong in the cable before being too weak to matter.

And for DC-driven signals, cross-talk is irrelevant. Cross-talk doesn't happen when you have a fixed current/voltage in the wire - without change, you can't induce currents in the neighbor wires.

The things you should think about here is:
1) What state will the ULN2003 (and your relays) take when you disconnect the ribbon cable? You want a signal failure to result in the relays enter a safe state.

2) Same thing when your processor is in reset state.

3) Same thing if your system is only partially powered - power to the relays and ULN2003, but not power to the microprocessor side.

The ULN2003 is just a darlington array. But in case you insert extra logic on the remote side of the ribbon cable, you may need to add series resistors as current limiters in case only the microprocessor side is powered - many logic chips will look like short circuits on the inputs when not powered - the input voltages gets clamped to VCC. A series resistor also have the advantage that it slows down the transition speed, reducing the amount of switching noise.