Preheat oven to 350 degrees. Butter a 9 x 9-inch baking dish and line with parchment paper, butter parchment too.
CRUMB LAYERS: In a large mixing bowl, whisk together flour, baking soda and salt. Add oats, brown sugar, and granulated sugar; whisk to blend. Whisk together melted butter and vanilla, then pour over oat mixture. Stir with a wood spoon until mixture is evenly moistened. Press 2/3 of the mixture into prepared baking dish, set aside remaining 1/3 crumb mixture.
FILLING: In a medium bowl, combine diced strawberries and rhubarb. Toss with lemon juice.
In a small mixing bowl whisk together sugar and cornstarch. Pour sugar mixture over strawberry mixture and toss to coat. Pour strawberry mixture over bottom crumb layer in baking dish. Evenly sprinkle remaining 1/3 crumb mixture over top.
Bake until the top is golden brown and crisp, and filling is bubbling, about 45 to 50 minutes. Cool on a wire rack, then cut into squares (I find it easier to refrigerate for an hour then cut into bars). Serve with vanilla ice cream if desired.