There is a date code embedded in the serial number of most washers and dryers. The charts in the image below show the serial number formats for the major manufacturers (Whirlpool, Frigidaire and GE). Kenmore products beginning with the prefix 110. . . are made by Whirlpool. Kenmore products beginning with 417. . . are made by Frigidaire. You can refer to the appropriate serial number format shown below to determine the date of manufacture for most Laundry appliances. You can also ask another question if you find an appliance that you like and need more information. We try to answer questions with 48 hours and can sometimes answer them sooner.
If you have around $300 to spend on the washer, I recommend that you consider the Kenmore model 110.2942 that is only $339. Here is a Sears.com link for that model of washer:
Kenmore White 400 3.2 cu. ft. Top-Load Washing Machine (2942)
. If this is out of your price range, I recommend that you look at the Kenmore and Whirlpool used washers that are similar to this model. This is a top load washer that has a dependable direct drive transmission. It has been manufactured for many years with relatively few problems. You may also be able to find a good deal on GE top load washers that begin with WDSR . . . (i.e. WDSR2120JWW). These types of washers are not the high efficiency type but they are dependable and likely a good buy as a used appliance. If the washer is older than 5 years, I wouldn't expect to pay more than $100 for it.
If you are looking at used front load washers, I recommend extreme caution when buying this type of washer used. They are expensive to fix. The repair parts are usually high priced. There is not likely a good buy available on this type of washer at this time. A new type of direct drive front load washer from Samsung, LG and Kenmore (796. prefix) has emerged in the last couple of years. This type of direct drive front load washer is not likely available on the used market at this time. This type is more dependable and longer lasting than any of the belt drive models that are currently available as used models. I recommend that you search for a top load washer since you are buying used at this time.
Kenmore dryers that begin with the 110 prefix and Whirlpool dryers are the type that I recommend shopping for. Dryers tend to last a long time and are simple to fix when the stop heating or break.
I recommend that you check Craigs List and other online forums for laundry appliances in your area. Be sure that you physically examine the product if possible before buying it.
Resubmit your question with details of the product that you are considering if you need more help.