Does MSSQL Server need more space than the size of the data itself for importing? How much more?
The error messages are indicating that the log file is full. SQL Server has two files (at least) for every database; one is the data file and one is the log file. On your particular server, it is possible that the log file is on a disk that doesn't have enough space for the log file while the data file may be on a disk where it has plenty of space.
If the database is in Simple recovery mode, simply execute a full backup and the log file will clear. However, you may need to find out where the log file is located and make sure the disk has enough space to house the log file while performing the BCP. You can do this by right-clicking on the database and going to Properties. Then click on Files on the left side. This will show you the physical path for the log file (.ldf).