A freezer is a better choice than a fridge.
Use a glass bottle and leave a little headspace. In theory cobalt glass is the best but it probably doesn't matter. Place the bottle within another waterproof container such as an ice cream box or similar, or just double-bag it within ziplock bags. Put it in the coldest part of the freezer. E-liquid won't freeze because it has a very low freezing point, but the low temperature stops decomposition.
Unflavored liquid will theoretically keep longest but even flavored liquid should be OK for 3 years plus.
Don't use a fridge, especially with plastic bottles. There is something about the damp environment that transfers flavors between containers, and allows materials to decompose. Water vapor gets in anywhere via osmosis. Either use a freezer or a cool, dark cupboard above child reach height. Unflavored liquid should have a shelf life of 2 years. Flavored liquids are another matter, it depends entirely on what types.