Yes they can start a partnership firm.

The procedure is get the partnership deed prepared and make it on a stamp paper of Rs. 500/-.

If you want a registered firm then you need to get it done through court and its procedure as under:

Under Section 58 of the Act, a firm may be registered at any time ( not merely at the time of its formation but subsequently also ) by filing an application with the Registrar of Firms of the area in which any place of business of the firm is situated or proposed to be situated. 

• Application shall contain:-
  • name of the firm
  • place or principal place of business
  • names of any other places where the firm carries on business.
  • date on which each partner joined the firm
  • name in full and permanent address of partners.
  • duration of the firm 
     
• Application shall be signed and verified by all the partners or their duly authorized agents.
   
• Application shall be accompanied by prescribed fee as well as the following documents:
  • Prescribed Registration Form for Incorporation of a Company. (Form No. 1 and Specimen of Affidavit)
  • certified true copy of the Partnership deed entered into.
  • ownership proof of the principal place of business
     
• Name of the firm should not contain any words which may express or imply the approval or patronage of the government except where the government has given its written consent for the use of such words as part of the firms name.

After getting the firm registered, it will perform as like as a compnay, each and every details / changes are to be submitted to registrar.

Since the partners are husband and wife, I am of the opinion not to get the firm registered.

Partner are husband and wife then why firm should not to be registered..??

Effects of Non-Registration:-

1. An unregistered firm cannot enforce its rights against the third parties.
2. The partners of an unregistered firm cannot sue its fellow partners or the firm for the enforcement of their rights.
3. Partners of the unregistered firm cannot be sued by the firm for exercising its claims.
4. Unregistration of a firm does not affect the rights of the third party to sue the firm for the recovery of dues.
5. Adjustment regarding the amount receivable or payable cannot be made.

Non-Effects of Non-Registration:-

1. An unregistered firm can file a suit for not exceeding Rs.100.
2. Powers of an official assignee or receiver remains unaffected.
3. Despite of unregistration of the firm, partners can still file a suit for the dissolution of the firm.
4. Partners can file a suit for the settlement of accounts.
5. Unregistration of firm does not affect the right to realize the property of the firm.
6. Adjustment regarding the amount receivable or payable can be made upto the amount not exceeding Rs.100.
7. No effect on the rights of the firm and its partners having no place of business in India.
8. No effect on the right to realise the damages by the dissolved firm for breach of a contract.
9. No effect on the partner’s right to refer the dispute to arbitration.
10. An unregistered firm can be sued by the third party.

Now the decision is your's to make.