What does ...原来档案都要统统装进铁皮柜，从头干起... mean? I can't really catch the meaning of this sentence. It was only given as an example of the use of 统统， I have no more context.

无论是谁，到新公司后，原来档案都要统统装进铁皮柜，从头干起。

No matter who, after arriving at a new firm, (your ) original files all (will) all be stored in a steel locker, you (can/must/should) start anew.

Or is '铁皮柜‘ some kind of slang for '计算机‘？？

What are '原来档案‘？？
'铁皮柜‘ is a metal filing cabinet.

'统统' is for emphasis. In this case, it is ALL the files, with no exception. 

Compare these two sentences, which are identical except for '统统'. Their meanings are basically the same, but the first one is more emphatic.

无论是谁，到新公司后，原来档案都要统统装进铁皮柜，从头干起。

无论是谁，到新公司后，原来档案都要装进铁皮柜，从头干起。

No matter who it is, when one enters the new company, one starts anew, with one's original files (hinting past history) stored away. 

There is only one subject in this sentence, it's the one who enters the new company, the one whose files are put away, and the one who has to start anew.