Restore in Full Recovery model with no transaction log backups
Yes, you should be able to restore and you'd be at the point in time of the backup.

Restoring from a backup will literally create a physically identical structure as existed at the point of the backup - data and log files will be the same size, so yes, you will still have a 300 GB log file if that was its size at the time of the backup (even if it was only 1% full).

Now, you need to do one of two things once you're back up and running:


Turn on transaction log backups to better protect your data in the event of another failure in the future.
Switch to simple recovery mode. Normally (there are always exceptions), this will prevent your transaction log from growing as much as it has. Your ability to restore remains the same as in full recovery and not bothering to take log backups.


For a lot more information about what you should do next (including what to do about your 300 GB log file), please take the time to read this question and its answers - yes, it's a lot of information, but it's going to be invaluable, I promise:

Why Does the Transaction Log Keep Growing or Run Out of Space?

Your gut reaction may be to try to shrink the log file to as small as possible, but that's not necessarily a good idea.