Ich komme ohne dem aus.

English Translation

I can do without this.

This sentence is incorrect. The accusative case must follow “ohne”.

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Technically you are right, but practically, this “mistake” is quite common. If you google “ohne dem” there are a lot of hits. Apparently, according to a stackexchange question I found, this is also common in some dialects, especially in Austria (Graz region).

Personally, it doesn’t sound wrong to me, but for me this only works if “dem” is alone and not followed by a noun. For some reason I would even prefer “dem” over “das” or “es” here.
I guess in this case it’s similar to the weird “dem” in the phrase “dem ist nicht so” (see also Sie können sagen, dass das keine große Rolle spielt. Dem ist jedoch nicht so.). But you do find lots of examples on google where a noun follows.

There is however a different case where “ohne dem” is absolutely correct, and that is “ohne + zu”, where the case is dictated by the verb:
Sie ist gegangen, ohne dem Mann zu helfen.