sure, its possible to even get a name brand for that budget in that size, provided you dont expect many features.
we got a samsung 1080p 50" un50eh6000fxza for about $800 a few years back. its a non smart tv with led backlighting. it has 120hz however we disable it.
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i'd suggest staying within samsung, sony and lg. if you can get grid-array backlighting go for it as contrast will be better than edge lit. i would not worry too much about smart functions (if you need them just get an external box to do it) as on-tv functions are generally poor, go out of date and are not updated often. i wouldnt worry too much about 120hz as well since you might end up disabling it since it makes video look odd (plasticky and fake) for anything but...