das Unternehmen
company
An economic organization that produces goods or offers services; a company or business.
Sie leitet ein mittelständisches Unternehmen mit 50 Mitarbeitern.
She runs a medium-sized company with 50 employees.

company / undertaking
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company
An economic organization that produces goods or offers services; a company or business.
Sie leitet ein mittelständisches Unternehmen mit 50 Mitarbeitern.
She runs a medium-sized company with 50 employees.

The noun 'das Unternehmen' (company/undertaking) is derived from the verb 'unternehmen' (to undertake), which means 'to do something' or 'to tackle something'. This origin explains both meanings: a company is a large project, and an undertaking is something that one undertakes.
Die Gründung einer eigenen Firma ist ein großes Unternehmen.
Starting your own company is a major undertaking.
To distinguish the meanings, pay attention to the context. If it is about business, employees, or products, it means the company (m01). If it is about plans, adventures, or risks, it means the undertaking (m02).
Das Unternehmen investiert in neue Technologien.
The company is investing in new technologies.
In a business context, 'Unternehmen' (company) is often combined with certain verbs. Typical phrases are: 'ein Unternehmen gründen' (to found/start a company), 'ein Unternehmen leiten' (to run a company), or 'für ein Unternehmen arbeiten' (to work for a company).
Nach dem Studium möchte er ein eigenes Unternehmen gründen.
After university, he wants to start his own company.
In the meaning of 'Vorhaben' (undertaking/project), the word 'Unternehmen' is often used in a somewhat more formal or narrative style. It usually describes a larger, planned action.
Die Rettungsaktion war ein schwieriges Unternehmen.
The rescue operation was a difficult undertaking.
house
'das Haus' and 'das Unternehmen' can both mean a company or institution. 'Unternehmen' is the normal, clear word for an economic organization. 'Haus' is used in this sense more in certain fields or in formal language, for example in publishing, banking, fashion, or hotels. It often sounds like 'our company' from the company's own perspective. In everyday language, 'Unternehmen' is usually easier to understand. 'Haus' fits when the style is a bit more official or when an industry uses this wording.

Wegen der hohen Kosten sucht das Unternehmen neue Partner.
Because of the high costs, the company is looking for new partners.
Unser Haus arbeitet seit vielen Jahren mit diesem Verlag zusammen.
Our firm has worked with this publisher for many years.