I currently have a 220 volt/30 amp service to an electric washer/dryer combination. The washer/dryer broke down and I will be switching to individual units each requiring a 120 volt/15-20 amp supply suggesting a separate circuit.

The current unit had a four wire configuration (2 hot - red/black, 1 neutral - white, and 1 ground - copper) with a double pole breaker. After removing the wire from the unit, I noticed that there was another wire (blue) which was not capped at the unit.

I know that I can replace the double pole breaker with 2 individual 20 amp breakers. Can I use the same wiring to hook up two 110 lines? I'm assuming that I would go red-white on one line and black-blue on the other but there is only one ground.

Let's see.
Problem #1. The colors: Easy. Use white and black eletrical tape to code the colors at both ends.

Problem #2. The 30 A wire isn't compitable with a 20 A rated outlet. Fix. Pigtail a 12 AWG wire to the 30 A rate wire in the box.

Problem #3: Ground. You can place everything in the same box. Use a 4x4 (I think) and separate the tabs on the duplex outlet and wire them separately or put two duplex outlets in the 4x4 box. Use a deep box or an extender. Don't know what you have to work with.

Problem #4: Breaker - yes

Problem #5: Put a note by the panel what was done.

Potential problem #6: Exceeding box fill (# of wires allowed).

If you use a duplex receptacle and break the tab off, it will then be considered as a multi-wire branch circuit. You must use a 2 pole 20 amp breaker or use handle ties on individual breakers if you go this route.

The reason this is in the code is that people only check that one of the two receptacles is not powered before starting to work on them. The breakers being operated together prevents those types of assumptions from killing people.

You can break both tabs off, can you not?

Article 100 Definition
Branch-Circuit, Multiwire.
A branch circuit that consists of two or more ungrounded conductors that have a voltage between them, and a grounded conductor that has equal voltage between it and each ungrounded conductor of the circuit and that is connected to the neutral or grounded conductor of the system.

Simply breaking both tabs does not change it's status as a multiwire branch circuit and that type of branch circuit is covered here..

210.4 Multiwire Branch Circuits
(B) Devices or Equipment
Where a multiwire branch circuit supplies more than one device or equipment on the same yoke, a means shall be provided to disconnect simultaneously all ungrounded conductors supplying those devices or equipment at the point where the branch circuit originates.

You can share the neutral on this receptacle... just leave the tab in place on the neutral side, and break it off on the hot side. You don't need more than three wires for this installation. And tape (cap) the blue wire at both ends.