These things depend a lot. Consumption has more to do with your modeling than with the database, and the way you model one is different from modeling another.
Contrary to what people imagine, Mongodb or any tool does not work miracles. My perception is that it will be worse because it is optimized to work with documents, expected from it large entries and without a pattern, so each document must have metadata to "explain" what that is. But without testing and measuring the specific case I can’t say.
Relational databases are highly optimized for space consumption and processing, and current implementations are very flexible and easy to control transactions in. Your disability is basically in more complicated access with need to make relationships (but it brings advantages too) and greater difficulty of vertical scale (but that is less necessary than in other solutions).
You need to see ways to reduce that data, see if you’re using the right types, the right way, if you can do some sort of compression. Mongodb does some compression by default, but only because he wastes too much resources not to. If compression helps then do it in SQL Server, since this is not standard.
If data enters the database endlessly in any technology it will take up a lot of space and there is no miraculous solution to change that. It is possible to do hard engineering work to find solutions.
In general lines this, can not help more without knowing the problem more deeply.
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