Where is the effectiveness of Access to handle large data ?
Perhaps the biggest advantage is the ability to have multiple concurrent uses whereas Excel limit is one. Second, a single table made from an Excel dump will likely result in one of the most inefficient db designs you can create as it will no doubt totally lack any resemblance of normalization. Access will surely consume far more space when replicating a spreadsheet design (wide, many fields) vs tall (many rows but in related tables). I would recommend you study db normalization and when you understand it, attempt to replicate your data with a properly normalized design - or stick with Excel.
As for approaching the size limit, one could look at mySql or Sql Server or similar as a back end option, but they are less forgiving when it comes to poor design.
The more we hear silence, the more we begin to think about our value in this universe.
Paraphrase of Professor Brian Cox.