It could be because bigger than needed because WordPress keeps in it some old stuff like post revisions, autodrafts, orphaned or duplicated meta… this could be easily cleaned using WP-Sweep plugin as long as you can run it on your site. I’m not sure how is your situation right now, are you able to use wp-admin or not. Anyway:
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As a first step I’d recommend backing up your database using PHPmyAdmin or much smaller alternative – PHPminiAdmin. With any of those tools you can also see the size of the tables, so you can check if the hosting company doesn’t lie to you. Notice that besides the table size, for each table there’s and index which depending on the table structure can take a reasonable space too.
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a) If you have access to the wp-admin – install the plugin and do some cleaning – the options are quite self-explanatory.
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b) If you don’t have wp-admin access and you don’t want to pay for an upgrade I think you should download the database and the files and setup your site somewhere else just for the sake of running the plugin and cleaning the database. Many hosting companies offer a free trial period. Alternatively you can set up the site on you local machine using XAMP server setup for instance and do the same. Having cleaned up the DB you can upload it back to the server.
From my experience sometimes the WordPress database can take many times more space than really needed, because of the trash collected with the years of working, so you could win this situation not paying for the upgrade.