Yes you can: many outlying rural properties in Australia not connected to reticulated water have been using such a such a system for many years. It requires: -A washing machine that operates on LOW pressure (most are high-pressure designed for reticulated systems). Ensure you read its manual/instructions carefully: machines differ.
-a tank on a stand or hill sufficiently high for gravity feeding (aim for 20 feet at least: the greater the height the greater the pressure.)
-a tank of sufficient capacity (eg tank of 1000 gallons/4500 litres ) depending on the usage.
-A pump to get it up there in the first place.
See the link below.
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