First, if you experience respiratory problems for any reason, it is always a good idea to seek medical assistance. You are encouraged to do so.
Skunk spray contains a number of volatile chemicals. One found in the secretions of striped skunks is 3-methyl-1-butanethiol. The Material Safety Data Sheet (MSDS) for this chemical states:
"Potential Health Effects
Eye: Causes eye irritation. May cause chemical conjunctivitis and corneal damage.
Skin: Causes skin irritation. May cause cyanosis of the extremities.
Ingestion: May cause gastrointestinal irritation with nausea, vomiting and diarrhea. Ingestion of large amounts may cause CNS depression.
Inhalation: Causes respiratory tract irritation. Vapors may cause dizziness or suffocation. Can produce delayed pulmonary edema. May cause burning sensation in the chest.
Chronic: Prolonged or repeated skin contact may cause dermatitis. Effects may be delayed.
Section 4 - First Aid Measures
Eyes: Flush eyes with plenty of water for at least 15 minutes, occasionally lifting the upper and lower eyelids. Get medical aid.
Skin: Get medical aid. Flush skin with plenty of water for at least 15 minutes while removing contaminated clothing and shoes. Wash clothing before reuse.
Ingestion: Do not induce vomiting. If victim is conscious and alert, give 2-4 cupfuls of milk or water. Never give anything by mouth to an unconscious person. Get medical aid.
Inhalation: Remove from exposure and move to fresh air immediately. If not breathing, give artificial respiration. If breathing is difficult, give oxygen. Get medical aid.
Notes to Physician: Treat symptomatically and supportively."
Source MSDS Creation Date: 9/02/1997, Revision #6 Date: 10/03/2005
There are additional nasty chemicals found in skunk spray. See FAQ 1032 for more information on deodorizing strategies. The web site by Professor William Wood skunk deodorizing has additional information.