After sealing a 325-run win over Nottinghamshire on Thursday, Somerset needed Middlesex and Yorkshire to draw at Lord's for them to become champions.
But Toby Roland-Jones' hat-trick sealed a dramatic 61-run win for Middlesex and their first title since 1993.
"A lot of us couldn't watch. It was an emotional, up-and-down rollercoaster," batsman Hildreth told BBC Somerset.
Spinner Jack Leach added: "It was a nightmare. A lot of us were struggling to watch it."
Somerset finished four points behind Middlesex, finishing second in Division One for a third time since 2010.
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Captain Chris Rogers - who announced his retirement on Thursday after hitting two centuries in his final match - said: "To come second and nearly win, it's a pretty special achievement.
"To go to Headingley and thump Yorkshire shows what a good side Somerset is. I think there are some special times ahead.
"You have got a lot of players who are still two or three years away from becoming very consistent players, in terms of their development."
Hildreth, who fractured his ankle while batting on day one against Notts but went on to make a hundred, continued: "There are such good signs from this year. We have such a good crop of young talent.
"If you can build that momentum over the next few years, it could be a great team."
Opener Tom Abell added: "We're all pretty gutted in the changing room. We came so close. To have it taken away in the final moment of the season is very disappointing.
"At the start of the season we were written off a little bit. The end of the season probably came at the wrong time.
"It's a great advert for county cricket, what's happened over the last week or so and how it's unfolded."
Leach, who ended the season with 65 County Championship wickets after taking 4-69 to help beat Nottinghamshire, revealed he has signed a new three-year contract with Somerset.
"I can definitely keep improving. I'm excited that I can keep getting better," said the 25-year-old. "I do see myself staying here for a long time.
"We've done well this season and we've come up with a method to win at home now, which is massive for us."
Watching on as Somerset were denied a first County Championship title was "horrible", says James Hildreth.