The first thing that came to mind was, if your worried about tacky, then why did you get a two year old. The second thing was, I bet you will wonder why you even asked this question a year from now. :)

if you are not going to be helpful, dont respond!!!!

Couple more questions.......Do you already have a carpet to put down or would you be buying it? If you would be buying it, then don't throw money on a solution that is temporary. Can you lift the carpet and install laminate or wood plank in that area.instead? If you already have an area carpet, the success of this depends on how thick the wall to wall carpet is. If it is a low pile or a loop, will work better. Also, how deep is the pile on the area rug? You will still have a cleaning issue, so it's not the best choice overall. Would be good to know which climate you live in too. Maybe tile is your best option, I'm not sure.

WOULD LOVE WOOD OR TILE FLOOR BUT its not in our budget and we just moved in and the carpert is brand new, it flows from the living to ding area, we live in a hot dry area in California

I live in a hot dry area of California too, and it's supposed to get even hotter this weekend! Have you consider pulling up the carpet and installing a plywood floor? Before you dismiss it, check out the plywood floors in this ideabook, they're pretty cool: https://www.houzz.com/magazine/try-diy-plywood-flooring-for-high-gloss-low-cost-stsetivw-vs~8216171

Get a throw rug that's washable for under the chair (or high chair) that your child sits in at the table.

Use an indoor/outdoor rug. They're easy to clean and the styles and textures are wonderful now. Check out home decorators or grandin road, also ballards

I like the look of layered rugs although I know some don't. Here is a visual for you. The indoor outdoor rug is a good idea. However as mentioned above depending on the thickness of your wall to wall, it may not look right as the furniture would create dents and folds in the rug. I have heard this can be prevented with a good rug pad.
If the layering doesn't work, they do have those disposable under high chair things but I've never used one.
http://houzz.com/photos/93201

Good call sam0705, I didn't even consider that the rug might be plush. Naughty me :)

I am having that problem as we speak as I try to protect a light rug from my kiddos.

You could always keep a shower curtain liner or even a large plastic contractor bag nearby and just place under chair while feeding two year old and whip it away again after meal. Low cost option and no one is the wiser :)