A lot of Honda Odysseys seem to have issues with the transmission. It seems to impact the 1999-2004 models the most. When that happens you have a few options with the first being to check if it is still under a warranty. Beyond that you take the path of rebuilding it yourself or getting it rebuilt. I would not suggest driving the van once it is in this state. It should only be driven to get you to a destination for repair. It does have the potential to lock up on you if you continue to drive it with these issues.
The 4-speed automatic transmission in 1999 to 2001 models had serious problems with transmission durability. Honda spokesman Mike Spencer stated that four-speed models were afflicted with a bad bearing that could break apart, scattering fragments of metal that clogged fluid passageways in the transmission, causing it to shift erratically. Honda responded to the problems by extending the warranty on the transmission on U.S.A. ’99-’01 models to 7 years or 100,000 miles (160,900 km). A class action settlement further extended coverage to 109,000 miles (175,400 km) or 93 months for some ’99-’01 Odysseys in the U.S.A.
(excerpt from Wikipedia)
Here are few resources that may help as you decide the best path for you:
Purchase Transmission ($3,000-$6,000): The most common route with this problem is to purchase a rebuilt transmission. You can purchase one from a dealership or a local shop. A Honda dealership will quote you from $4,000-$6,000 on a rebuild. A local mom/pop shop will be around $3,000-$4,500 for a rebuild.
DIY Rebuild Transmission ($1,000-$1,500): There are people who have done this successfully and have posted about the experience here and here. (These posts cover the steps, parts, cost, good and bad of this repair.) I would say that if you are not one who is already good with car repair or engine work then you should pass on this, but it is an option. This option will cost around $1,000 as documented here.
This video will give you a better idea of what is involved in a DIY rebuild: