As a regular user of the A303 (it runs within two miles of here), I always wince when I see people cycling on it. There's at least one chap who seems to commute on a fairly ordinary looking bike (I don't mean an 'ordinary'; that
would be silly) and, while I wouldn't do it, he's been at it for some time. Round us there are parallel roads for most of the way, but they're not particularly nice either, so it might be the least evil way to do a commute.
Of course, the real problem is not that the A303 is too busy, but that it's not busy enough.
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the 100 mile parking lot that is the A303).
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The 303 is a perfectly good road to drive during the week. It's not particularly good if you are going with the flow at a weekend (westbound on Friday/Saturday; eastbound on Sunday afternoon). It is hell at weekends during the main holiday season. I was doing a weekly commute using it last year. There were three occasions when I was delayed, once by ten minutes (wagon broken down on a single carriage way section), once when I had to come A31/A35 instead as the 303 was completely closed (on a dualled section) and once for the summer solstice. Monday mornings going eastbound, never had a problem.
It's always been marginal as to whether it's worth upgrading as the level of business travel isn't that high. And that's before we get onto the 'what shall we do about Stonehenge' question. It is the last 'unfinished' major road in the south, but I fear it is likely to remain so.
But Yeovil CC do use it for time trials.