If one was worth 5,000 a year in Jane Austen's time, what would that be in American dollars?

In 1988, it would have been about $165,000. Here's a chart with conversions for Jane Austen's novels:

web.stanford.edu/~steener/su02/english132/conversions.htm webpage no longer exists

The Telegraph updated the 1988 calculations in the 2015 article linked below.

According to the Telegraph, using 2013 figures, 5,000 pounds a year (the income of Mr. Bingley in "Pride and Prejudice") would have a real value of 309,550 pounds in 2013 (or, using the current conversion ratio of approximately 1.50 US dollar for 1 UK pound, $464,325 in US dollars).

However, this simple conversion does not take into account the drastic changes in cost of living since Austen's day. Basically, that income in Austen's day, if adjusted for the changes in cost of living as well as inflation, would produce the relative wealth (which the Telegraph calls "prestige value") of an income of 4,438,500 UK pounds ($6,657,750 US) in 2013. That's a fairer reflection of how Mr. Bingley would live today -- and Mr. Darcy had twice Mr. Bingley's income.

http://www.telegraph.co.uk/culture/books/11063670/Could-Mr-Darcy-afford-a-stately-home-today.html

Yes. According to this site £5,000 in 1814 would be £305,200 (about $457,800) in 2014 using the cost of living. However, if calculated as a proportion of the GDP (total output) it would be a staggering £16.17 million (about $24.255 million) - and the two sites are close on the "prestige value" (£4.822 million or $7.233 million). Exchange rate used: £1 = US$1.50

https://www.measuringworth.com/calculators/ppoweruk/