Does MSSQL Server need more space than the size of the data itself for importing? How much more? Does MSSQL Server need more space than the size of the data itself for importing? How much more? How to tell?

Recently tried copying data into MSSQL server (2012)

Microsoft SQL Server Management Studio          11.0.3156.0
Microsoft Analysis Services Client Tools        11.0.3130.0
Microsoft Data Access Components (MDAC)         6.2.9200.16384
Microsoft MSXML                                 3.0 5.0 6.0 
Microsoft Internet Explorer                     9.10.9200.22387
Microsoft .NET Framework                        4.0.30319.18449
Operating System                                6.2.9200


using BCP and was seeing errors like


  Error = [Microsoft][ODBC Driver17 for SQL Server][SQL Server]The transaction log for database 'EpicShd' is full due to 'CHECKPOINT'.


and


  Error = [Microsoft][ODBC Driver 17 for SQL Server][SQL Server]The transaction log for database 'EpicShd' is full due to 'ACTIVE_TRANSACTION'.


When looking at the DB itself, it threw...


  Database cannot be opened due to inaccessible files or insufficient memory or disk space.  See the SQL Server errorlog for details. (Microsoft SQL Server, Error: 945)


For context, the total size of the data (at least the TSVs that BCP was copying in) was ~10G, while the total storage space on the DB server was 28G, so I would think there would not be a problem. I am bringing in multiple tables each being 500MB-1GB and doing it in set of 1-2 at a time.

Does copying data into MSSQL server take more space than the data itself? How much (temporary storage)? How can I tell (not much experience with DBA)?
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).