I saw a job posting that requires Active DOD Secret Clearance. I was wondering if there are any lifestyle restrictions if I were to work for that company. I like travelling internationally and wonder if I would not be allowed to anymore.
Just for clarity's sake, DoD Secret is not a Top Secret clearance. Those are separate levels of restricted access. Additionally, requiring an "active" clearance means you must already be cleared for classified work. Most companies can hire someone without a clearance, if it's obvious they will be able to obtain one, but they will almost always preferentially hire someone who is already cleared.
To begin, you would have to fill out SF-86 (standard form 86), which can be found here. Note that an average citizen cannot just fill out this form and get a clearance. You must be sponsored by a company with active classified contracts.
The SF-86 will require you to answer many questions such as 10 years of employment and residency information, all historical police records, any use of drugs, any previous criminal charges involving firearms, alcohol, drugs, or other felonies, whether or not you've ever advocated for overthrow of the US government, etc., etc.
You will also be required to disclose any mental health issues which may affect your "judgment, reliability, or trustworthiness even if you are not experiencing such symptoms today."
You will also have to provide numerous witnesses for each category within the form, and these witnesses will be contacted and interviewed about your "judgement, reliability, and trustworthiness." Things which may make it difficult to receive a clearance which haven't already been mentioned include (but are not limited to) extra-marital affairs, abnormal sexual fetishes, contacts with foreign nationals (primarily only if they're from countries which the US is not on good terms with), history of bad credit, and poor debt to income ratio. Basically, if the US government thinks you can be easily black-mailed, you'll be considered ineligible for a clearance.
So are there any lifestyle restrictions?
Yes, if any of the above applies to you. Also yes, even if they don't. In the case of your example about foreign travel, all foreign travel must be disclosed to the FSO (facility security officer) several months in advance. They will then brief you on security risks in the region you're traveling to (the federal travel advisory map can be found here) and you are not permitted to travel to certain countries (hopefully you weren't planning on taking a trip to Iran or China anytime soon, anyway).
Also, if you smoke marijuana, even if it's legal in your state, that makes you ineligible for a clearance, as it's still illegal at the federal level. The DoD tends to have more leniency here, and you'll be eligible if you can show that you haven't smoked marijuana within a few years, but YMMV. There has also been DoD guidance that use of CBD oil must be disclosed as well.
Note that any of the items which can make you ineligible for a clearance can also cause you to lose your clearance (and job) if they come up after you're already cleared. You're also required to undergo a re-investigation every so often (I believe it's 10 years for Secret, 5 years for TS)
There's also one tidbit to remember: the SF-86 is a lifetime contract. Receiving a clearance means you've sworn to uphold the national security of the US for the rest of your life, even if you change citizenship. This even includes going to websites like WikiLeaks - if you see classified information in a non-cleared environment, you are required by law to disclose it, even if you no longer have an active clearance.
I've held clearance in the past. "Secret" clearance is not terribly hard to get and comes with no general restrictions. You're expected to be financially and personally responsible, and you're expected to report suspicious contacts (e.g. people aggressively asking for secret details about your projects/activities).
Secret clearance will not impact any of your travel activities. It is not until you get into Top Secret/SCI access that anything begins to impact your travel. Even that level of access typically does not significantly impact travel specifically, but there will likely be certain areas/countries where it might raise questions during an audit. As with any clearance, you will be expected to respond truthfully and report anything suspicious during your travels there.
When, very early in my career, I was offered a "secret project" by a college professor which would have required such a Clearance, I thought about it very carefully, then politely declined.
I've never regretted that decision. I think that there are plenty of good-paying jobs out there which "I can legally talk about," and I managed to find them in sufficient number. Yes, I fully understand the need for the restrictions that go with access to government secrets of any sort, but I declined to personally subject myself (and my family, and my neighbors, and my dogcatcher ...) to them. To me, "the cost/benefit ratio" simply wasn't there – and, "being quite-necessarily locked to a government pay scale" didn't appeal to me too much, either.
And, financially, "I did pretty well." 🤷♂️
The final decision is yours alone.
Have you considered reaching out to the company's HR department or sending an email to the Department of Defense?
I mean, we could try doing some googling (maybe someone here has got Active DOD Secret Clearance, but I'd suspect that's not going to be the case) - but would you trust that googled information over asking the source?
Both of those groups have a vested interest in giving you good solid information on the question. So why not reach out and ask them?