You're going to have to either get a washer that accepts 240V, or change the circuit.

New Washer

Getting a new washer will depend on the manufacturer, dealer, and other factors that are off topic here.

Change the circuit

Modifying the circuit will require knowledge of electrical safety practices, and general electrical wiring principles/standards/codes. If you lack this knowledge, or are not comfortable working with electricity, please contact a local licensed Electrician.

If you decide to tackle this project yourself, here is an overview of what needs to be done. You'll be going from a configuration with two ungrounded (hot) conductors, and one grounding conductor (NEMA 6-20). To a configuration with one ungrounded (hot) conductor, one grounded (neutral) conductor, and one grounding conductor (NEMA 5-20). To do this you'll have to swap out the double pole breaker for a single pole breaker, and repurpose one of the conductors.

At the Panel

1. Before you begin, you'll need to purchase a 20A single pole circuit breaker that is compatible with your service panel, and a blank panel cover insert.
2. Start by turning the power off by flipping the main breaker to the OFF position, then verifying that power is off using a non-contact voltage tester.
3. Remove the service panel cover (be careful working in the open panel, the main lugs are still energized even when the main breaker is in the OFF position).
4. Locate the breaker for the washer.
5. Remove the wires connected to the breaker (should be either black & red, or black & white).
6. Remove the breaker from the panel.
7. Install the new single pole breaker in one of the slots where the old double pole breaker was.
8. Install the blank panel cover insert in the panel cover, so that is covers the empty space left.
9. Connect the black wire that you removed from the breaker in step 5, to the new breaker.
10. If the other wire from step 5 is white, connect it to the grounded (neutral) bus bar. If the wire is red, mark the wire with a bit of white tape, or a white marker, then attach it to the bus bar.
11. Reinstall the panel cover.

At the receptacle

1. Before you begin, you'll have to purchase a 125V 20A receptacle.
2. Uninstall the old receptacle.
3. Connect the black wire to the brass screw terminal on the new receptacle.
4. If the other wire is white, attach it to the silver screw terminal on the new receptacle. If the other wire is red, mark it with white tape or a white marker, and attach it to the silver screw terminal on the new receptacle.
5. Attach the bare or green grounding wire to the green grounding screw on the new receptacle.
6. Mount the receptacle in the box, and install the cover plate.

Once everything is finished, you can flip the main breaker, and the new breaker to the ON position.

WARNING:
This answer is only applicable in the US, and possibly Canada.

You can use a 3 wire feeder to supply a separate building, if...

• The installation was in compliance with a previous edition of National Electrical Code (existing premises wiring).
• An equipment grounding conductor is not run with the supply to the structure.
• There are no continuous metallic paths bonded to the grounding system in each structure (bonded water, or gas piping, other conduit, etc.).
• Ground-fault protection of equipment has not been installed on the supply side of the feeders.

National Electrical Code 2014

Article 250 Grounding and Bonding

II. System Grounding

250.32 Buildings or Structures Supplied by a Feeder(s) or Branch Circuit(s).

(B) Grounded Systems.

(1) Supplied by a Feeder or Branch Circuit. An equipment grounding conductor as described in 250.118 shall be run with the supply conductors and be connected to the building or structure disconnecting means and to the grounding electrode(s). The equipment grounding conductor shall be used for grounding or bonding of equipment, structures, or frames required to be grounded or bonded. The equipment grounding conductor shall be sized in accordance with 250.122. Any installed grounded conductor shall not be connected to the equipment grounding conductor or to the grounding electrode(s).

Exception No 1: For installations made in compliance with previous editions of this Code that permitted such connection, the grounded conductor run with the supply to the building or structure shall be permitted to serve as the ground-fault return path if all of the following requirements continue to be met:

(1) An equipment grounding conductor is not run with the supply to the building or structure.

(2) There are no continuous metallic paths bonded to the grounding system in each building or structure involved.

(3) Ground-fault protection of equipment has not been installed on the supply side of the feeder(s).

If the grounded conductor is used for grounding in accordance with the provision of this exception, the size of the grounded conductor shall not be smaller than the larger of either of the following:

(1) That required by 220.61

(2) That required by 250.122

Changing from a 30A breaker to a 50A breaker can only be done, if you also change the wires to 6 AWG. In which case you'll have to follow current codes, and install 6/3 with ground. Breakers (and fuses) are always sized to protect the wire connected to them, so you can't change the breaker size without also changing the wire size (unless you're going down e.g. 50A to 30A).

However, depending on what you're doing, you may not have to change the breaker at all. If the planned circuits in the structure are not going to be fully loaded, you may well be able to supply the subpanel with a 30A breaker. Just because the subpanel has 50 amperes worth of overcurrent protection, does not mean the supply breaker has to be 50A. Whether or not you actually need a 50A breaker on the supply, depends entirely on what the subpanel will be powering.