Having your period is part of being a woman and completely normal.
There is nothing to be ashamed of.
If you feel uncomfortable taking out single pads from your bag bring a small purse that holds your pads in your bag.
You can also consider placing a box of pads in the bathroom with a sign that reads Emergency stock.
Help yourself and consider filling it up if empty or something like this for all women in the office space.
I bet some of them will be really thankfull if their period hits unprepared.
To include the suggestion from the comments, if there is no space available to put a box of pads, you could request a shelf or something like this from your management or facility management.
As a side note: there are ways to manage the flow that don't include pads that need to be changed regularly.
A cup just needs to be emptied once in a while (up to 8 - 12 hours if I remember correctly) and that can be done in the bathroom without carrying anything.
There are also period panties one can wear if the cup is not enough.
workplace.stackexchange.com/users/78026/Lehue
I think small metal boxes are available at most apothecaries to store and carry pads.
I use them and they are quite practical.
But, seriously, there's nothing embarrassing in having the period.
If a group of grown men are feeling awkward for this reason, it's definitely their problem, not yours
workplace.stackexchange.com/users/102625/schrodingercat
I personally use a small, decorative travel makeup bag that holds my sanitary supplies and some single-use packs of Advil.
I fill it at home and take the whole pouch with me to the restroom, so there's no smuggling supplies from one bag to another or into my hand.
In the fairly unlikely chance that someone even notices what I am carrying, it can easily be assumed that it's just to freshen up.
Even though I don't usually wear makeup at work, I doubt the men in the office overthink me sometimes entering the restroom with a travel makeup kit.
workplace.stackexchange.com/users/94856/Meg
Talk to your office manager to organize sanitary packages for the bathroom.
That should include things like disposable tooth brushes, antiseptic mouthwash, pads, toilet disposable wet naps (alternative to dry toilet paper), etc.
Having those things in the bathroom is not only a great convenience, but also a subtle way to remind everybody about hygiene.
Now, despite people telling you to "not be embarrassed," the reality is that the human brain is a bit more complex than that.
Trying to simply overcome something you're uncomfortable with doesn't happen by sheer brute force.
While I don't have to deal with it with pads, somewhere around spring and fall, I usually have a couple of weeks (perhaps due to temperature changes) where I have a perspiration problem.
Imagine wearing a t-shirt in the cold and still having sweaty armpits.
It can be really annoying and embarrassing.
If somebody tells me that I shouldn't be embarrassed about it because it's a normal human reaction, it's not just going to make it not embarrassing.
I have to take my medical-grade antiperspirant, walk myself to the bathroom and apply it out of sight.
So if you don't think you'll simply become the brave person that everybody says you should be (like I'm not), then feel free to adopt some strategies: Put it in your jean pocket; if you have a switcher with pockets, use those pockets.
Proceed to the bathroom.
workplace.stackexchange.com/users/111778/Kiril
You could treat them as what they are: Normal hygiene items.
Just put a Box on the toilet.
I know a lot of people doing that already, in the guest toilet, as well as restaurants/cafes having those available for their guests.
If you have female visitors/clients on site, they might appreciate it in a time of an emergency.
Get one of those larger handbags for work, put your personal stock in there as well as other toiletries you might need (comb, toothbrush etc.).
Make a habit out of it to carry this bag to the toilet with you.
workplace.stackexchange.com/users/71695/Daniel
How to take my pads with me to the toilet subtly?  I work in an all male office/open space and everytime I get my period I go through the awkwardness of trying to take them out of my bag and put them in my pocket an go to the toilet. I can't really take my whole bag, nor can I carry all the pads I need in my pants during the day.
How to take my pads with me to the toilet subtly?
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<work-environment><europe>
workplace.stackexchange.com/questions/149218
workplace.stackexchange.com/users/112449/Jenna
You could bring a small bag with some emergency supplies that you can carry to the toilet or leave a box in the bathroom. If you want to be subtle, you could simply get a small purse or make-up bag which you routinely take with you to the toilet. Alternatively, you could leave a supply in the toilet at all times.
You could acquire a purse or make-up bag that you get into the habit of always taking with you to the bathroom. You could keep a supply in the toilet.