Very doable, but a washer is a fair-sized load which would be somewhat above a 12V starter system. I know my front-loader consumes about 300W while spinning, and about 150 while washing, so let's assume a top loader uses about twice that, or 600W spinning and 300 washing?
Alternatively, you can figure out two different ways what the power consumption is. First, you can plug a killawatt unit into the wall, and the washer into that, and measure the total watts consumed. You could also get a clamp meter and measure the wattage while it's starting up, agitating, and finally spinning. Assuming a standard load runs for about an hour, I'd say total consumption is in the range 0.5 to 1.0kWh of power.
That is fairly large compared to
lights or a TV, that might only be consuming ~100W per hour of use.
My usual recommendation is to have 2X as many watts of panels as your load, so that would be in the range of 1000-1200W. Shop for grid-tie panels on Craigslist. Don't by 100W/12V automotive panels. They have the highest W/$ prices.
You will need the panels, a battery bank, a charge controller, and an inverter to convert the DC solar/battery power into AC.
Here is what I think you need.
1000W of panels ( about 260$ on Craigslist right now) Wire them 2S2P. That means two parallel strings of two panels in series.
50A charge controller (look at Epever's Tracer 5415AN, about 270$)
1500W sine-wave inverter (beware, cheaper inverters are square-wave, and burn out electrical motors quickly) Look at Samlex 24V/1500W for 550$
four 6V golf-cart batteries wired in series to get 24V (Costco has GC batteries for 99$ + tax and core-charge)
Keep in mind that a system big enough to power a washer is going to provide power for lights, TV, and a computer too, so although this is a rather substancial investment, you'll get a lot out of it.